Regular Series

Vol. 42 (2011), No. 5, pp. 893 – 1158

XXIII Marian Smoluchowski Symposium on Statistical Physics Random Matrices, Statistical Physics and Information Theory

Kraków, Poland; September 26–30, 2010

Non-Hermitian Extensions of Wishart Random Matrix Ensembles


We briefly review the solution of three ensembles of non-Hermitian random matrices generalizing the Wishart–Laguerre (also called chiral) ensembles. These generalizations are realized as Gaussian two-matrix models, where the complex eigenvalues of the product of the two independent rectangular matrices are sought, with the matrix elements of both matrices being either real, complex or quaternion real. We also present the more general case depending on a non-Hermiticity parameter, that allows us to interpolate between the corresponding three Hermitian Wishart ensembles with real eigenvalues and the maximally non-Hermitian case. All three symmetry classes are explicitly solved for finite matrix size \(N\times M\) for all complex eigenvalue correlations functions (and real or mixed correlations for real matrix elements). These are given in terms of the corresponding kernels built from orthogonal or skew-orthogonal Laguerre polynomials in the complex plane. We then present the corresponding three Bessel kernels in the complex plane in the microscopic large-\(N\) scaling limit at the origin, both at weak and strong non-Hermiticity with \(M-N\geq 0\) fixed.

CDT as a Scaling Limit of Matrix Models


It is shown that generalized CDT, the two-dimensional theory of quantum gravity, constructed as a scaling limit from so-called causal dynamical triangulations, can be obtained from a cubic matrix model. It involves taking a new scaling limit of matrix models, which is more natural from a classical point of view.

Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices — The Extended Version


This is a longer version of our article Burda et al., Phys. Rev. E82, 061114 (2010), containing more detailed explanations and providing pedagogical introductions to the methods we use. We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so-called \(M\)-transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues’ domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.

Asymmetric Random Matrices: What Do We Need Them For?


Complex systems are typically represented by large ensembles of observations. Correlation matrices provide an efficient formal framework to extract information from such multivariate ensembles and identify in a quantifiable way patterns of activity that are reproducible with statistically significant frequency compared to a reference chance probability, usually provided by random matrices as fundamental reference. The character of the problem and especially the symmetries involved must guide the choice of random matrices to be used for the definition of a baseline reference. For standard correlation matrices this is the Wishart ensemble of symmetric random matrices. The real world complexity however often shows asymmetric information flows and therefore more general correlation matrices are required to adequately capture the asymmetry. Here we first summarize the relevant theoretical concepts. We then present some examples of human brain activity where asymmetric time-lagged correlations are evident and hence highlight the need for further theoretical developments.

Theory of Stochastic Canonical Equations of Random Matrix Physics, SOS Law, Elliptical Galactic Law, Sand Clock Law and Heart Law, LIFE, Sombrero and Halloween Laws


Our studies are essentially based on the martingale differences method developed in my previous papers for resolvents of random matrices. This method possesses the self-averaging property of the entries of resolvents of random matrices and, hence, we can deduce the stochastic canonical equation. The lecture contains the most important results from numerous papers and books dealing with the theory of Unitary random matrices and functions of random matrices. We give the REFORM method of proving of all results, avoiding the method of moments. We do not try to describe here all known properties of the eigenvalues and eigenvectors for all classes of random matrices. However, our aim is rather to present the theory of stochastic canonical equations, and to give rigorous proofs of the procedures used to deduce these equations on the base of the author’s General Statistical Analysis. We consider special classes of analytic functions of random matrices. The description problem for normalized spectral functions of some analytic functions of random matrices is discussed in detail. Specifically, we present here the new theory: LIFE, which is the abbreviation for Limit Independence of Functions of Ensembles.

Doorway States Coupled to a Background: Fidelity and Survival Probability


The doorway mechanism in which a distinct state is coupled to a background is encountered in a rich variety of systems. Similar scenarios are likely to be relevant in quantum information theory. We review recent analytical and numerical results obtained for various statistical observables: the distribution of the maximum overlap coefficient between doorway and the true eigenstates of the total Hamiltonian, the averaged fidelity which equals survival probability, and the distribution of fidelity.

Complex Geography of the Internet Network


The geographic layout of the physical Internet inherently determines important network properties. In this paper, we analyze the spatial properties of the Internet topology. In particular, the distribution of the lengths of Internet links is presented — which was possible through spatial embedding of a representative set of IP addresses by applying a novel IP geolocalization service, called Spotter. The dataset is a result of a geographically dispersed topological discovery campaign. After showing the spatial likelihood of Internet nodes we present two approaches to describe the length distribution of the links. The resulting characterization reveals that the distribution can be separated into three characteristic distance ranges which can be mapped to the regional, transcontinental and intercontinental connections. These regimes follow a power-law function with different exponents.

Three-cycle Problem in the Logistic Map and Sharkovskii’s Theorem


In the logistic map a 3-cycle does not appear until after the end of stable \(2^k\) cycles. An impetus for analytical studies of 3-cycles is provided by Sharkovskii’s theorem, according to which the existence of a 3-cycle means the existence of all other cycles, hence chaos. It is a rigorous definition of chaos. We give a simple and direct proof of the existence of 3-cycles. The logistic map at fully developed chaos is shown to be isomorphic to the dynamics of a harmonic oscillator chain at the thermodynamic limit. Chaos in the logistic map is signified by a 3-cycle and in the harmonic oscillator chain by the thermodynamic limit.

Moments of Wishart–Laguerre and Jacobi Ensembles of Random Matrices: Application to the Quantum Transport Problem in Chaotic Cavities


We collect explicit and user-friendly expressions for one-point densities of the real eigenvalues \(\{\lambda _i\}\) of \(N\times N\) Wishart–Laguerre and Jacobi random matrices with orthogonal, unitary and symplectic symmetry. Using these formulae, we compute integer moments \(\tau _n=\langle \sum _{i=1}^N\lambda _i^n\rangle \) for all symmetry classes without any large \(N\) approximation. In particular, our results provide exact expressions for moments of transmission eigenvalues in chaotic cavities with time-reversal or spin-flip symmetry and supporting a finite and arbitrary number of electronic channels in the two incoming leads.

Tails of Composite Random Matrix Diagonals: The Case of the Wishart Inverse


We analytically compute the large-deviation probability of a diagonal matrix element of two cases of random matrices, namely \(\beta =\left [\mathbf {H}^\dagger \mathbf {H}\right ]^{-1}_{11}\) and \(\gamma =\left [\mathbf {I}_N+\rho \mathbf {H}^\dagger \mathbf {H}\right ]^{-1}_{11}\), where \(\mathbf {H}\) is a \(M\times N\) complex Gaussian matrix with independent entries and \(M\geq N\). These diagonal entries are related to the “signal to interference and noise ratio” (SINR) in multi-antenna communications. They depend not only on the eigenvalues but also on the corresponding eigenfunction weights, which we are able to evaluate on average constrained on the value of the SINR. We also show that beyond a lower and upper critical value of \(\beta \), \(\gamma \), the maximum and minimum eigenvalues, respectively, detach from the bulk. Responsible for this detachment is the fact that the corresponding eigenvalue weight becomes macroscopic (i.e. \(O(1)\)), and hence exerts a strong repulsion to the eigenvalue.

Composition of Quantum Operations and Products of Random Matrices


Spectral properties of evolution operators corresponding to random maps and quantized chaotic systems strongly interacting with an environment can be described by the ensemble of non-Hermitian random matrices from the real Ginibre ensemble. We analyze evolution operators \({\mit \Psi }={\mit \Psi }_s \dots {\mit \Psi }_1\) representing the composition of \(s\) random maps and demonstrate that their complex eigenvalues are asymptotically described by the law of Burda et al. obtained for a product of \(s\) independent random complex Ginibre matrices. Numerical data support the conjecture that the same results are applicable to characterize the distribution of eigenvalues of the \(s\)-th power of a random Ginibre matrix. Squared singular values of \({\mit \Psi }\) are shown to be described by the Fuss–Catalan distribution of the order of \(s\). Results obtained for products of random Ginibre matrices are also capable to describe the \(s\)-step evolution operator for a model deterministic dynamical system — a generalized quantum baker map subjected to strong interaction with an environment.

Random Matrices and Localization in the Quasispecies Theory


The quasispecies model of biological evolution for asexual organisms such as bacteria and viruses has attracted considerable attention of biological physicists. Many variants of the model have been proposed and subsequently solved using the methods of statistical physics. In this paper I will put forward important but yet overlooked relations between localization theory, random matrices, and the quasispecies model. These relations will help me to study the dynamics of this model. In particular, I will show that the distribution of times between evolutionary jumps in the genotype space follows a power law, in agreement with recent findings in the shell model — a simplified version of the quasispecies model.


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